= Certain ruin with nonpositive loading and finite claim variance
{title2=$c\le\lambda\mu\Longrightarrow\varphi(u)=0$}
For positive claims with finite variance and $c\le\lambda\mu$, the <classical risk model> has ruin probability one from any finite capital. At claim times, surplus increments are independent copies of $cT-X$ with mean $c/\lambda-\mu$. Negative mean sends their partial sums to minus infinity by the <strong law of large numbers>. At zero mean, the increments have finite nonzero variance. For every fixed $K$, the <central limit theorem> gives limiting probability $1/2$ of a partial sum below $-K$. The probability of unboundedness below is therefore at least $1/2$; as a tail event it has probability zero or one by the <Kolmogorov zero-one law>, and hence one.
Back to article page
